# Heuristic Optimization of a New Type of Prestressed Arched Truss

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions for post-tensioned concrete slab bridge decks, reporting that a cost increase of less than 1% reduces CO

_{2}emissions by more than 2%. In the literature [39,40,41,42,43], structural analyses, structural efficiency, and experimental study of prestressed truss systems were considered.

## 2. Problem Formulation

#### 2.1. General Aspects of Understanding the Prestressing Method

#### 2.2. Static Analysis of the Arched Trusses with a Cable

#### 2.3. Algorithm for the Calculation of Prestressed Trusses

#### 2.4. Disadvantages in the Calculation Method

#### 2.5. Disadvantages in the Structural Design Process of the Arched Truss with Tie Members (Cable)

## 3. Problem Solution

#### 3.1. Geometric, Physical and Mechanical Characteristics of a New Type of Prestressed Arched Truss

^{2}); elastic modulus (Young’s modulus) of the angular truss elements was: $E=\mathrm{21,000}$ (kN/cm

^{2}); design resistance of the cable was: ${R}_{u}=200$ (kN/cm

^{2}); and the elastic modulus (Young's modulus) of the cable was: $E=\mathrm{16,000}$ (kN/cm

^{2}).

#### 3.2. Problem Solution in the Structural Design Process Method

#### 3.3. Problem Solution in the Calculation Method

#### 3.4. Calculation of the New Model of the Prestressed Arched Truss Using the Matrix Stiffness Method and a Numerical Example

_{x}(1), u

_{y}(1) and u

_{y}(14).

#### 3.5. Optimization of Prestressed Truss

^{®}Core ™ i7 processor at 3.33 GHz.

#### 3.6. Sizing of the Cross-Section of the Truss Elements

## 4. Results and Analyses

- (1)
- A new geometric and innovative form was discerned for a prestressed arched truss that allows the development of higher values of prestressing force;
- (2)
- A new geometric and innovative truss was discerned that does not have unloading elements in the bottom chord, and the materials consumption exceeding problem was solved;
- (3)
- A new approach to the calculation (metaheuristic optimization) of a prestressed arched truss has been developed, resulting in the determination of the highest values of prestressing force with high accuracy in the tie member.
- (4)
- (5)
- By the new approaches offered by the authors, the weight of the truss was reduced, and the load capacity, rigidity, reliability, and fatigue resistance of the new truss were increased (see Figure 9);
- (6)
- The rate of prestressing force was increased (the asterisks are below the rings in Figure 9), as a result the elastic region of the working material was extended.

## 5. Conclusions

- It is recommended to use this new type of prestressed arched truss with a tie member as an innovative long-span roof system optimized and designed by the authors;
- It is recommended to adopt the innovative technique proposed in the process of designing the geometric form of a new type of prestressed arched truss. Thus, the tie member of the truss should be moved from the strut node towards the support node (Figure 2 and Figure 5), to eliminate unloaded elements in the truss structure and to develop a high rate of prestressing force in the primary structure of the truss;
- This research recommends using the performed approach in calculating prestressed arched trusses, where the highest values of the prestressing force are determined by the metaheuristic optimization and matrix stiffness methods;
- The research in this paper recommends the new approaches to obtain a truss with the lowest gravity, as the location of points of minimal weight of the prestressed arched truss by metaheuristic optimization were established and presented in Figure 8a,b;
- The optimization of the prestressed arched truss reduces the self-weight and increases the load capacity of the truss by 8–17%, depending on the span (see Figure 9);
- The new approach to increasing the rate of prestressing force of the truss achieves elastic region of working material expansion by almost 12–14%;
- It is recommended to use the new approaches, instead of several approaches for defining values of truss parameters (prestressing force in the cables, cross-sections in the truss elements), as far as it is possible to define them with a particular single approach;
- Reducing the weight of the new truss consequently reduces CO
_{2}emissions. - To obtain the prestressed arched truss with minimum weight is necessary to conduct future research not only to optimize the cross-sectional area, but also the truss topology (shape formation). In particular, it will be very relevant to optimize the shape of the chords, the boom lift, the type of lattice, and the number of panels of prestressed trusses.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Application of the prestressing technique in different structures: (

**a**) Ancient Egyptian prestressed ship; (

**b**) prestressing used for barrels; (

**c**) prestressed wheel.

**Figure 3.**Hierarchy of the structural efficiency of the arched truss. (

**a**) Non-prestressed truss; (

**b**) prestressed truss; (

**c**) loaded prestressed truss; 1—dotted line—initial position; 2—solid line—final position; (

**d**) graph comparing the prestressed trusses’ parameters to the non-prestressed trusses’ parameters.

**Figure 4.**Prestressed arched truss with tie member. (1) Prestressing of the truss by force $X$ at the I–I axis; (2) Loading of the truss by force $P$ at II–II axis; (3) Restoring of equilibrium by some force at the 0–0 axis (neutral axis).

**Figure 6.**The view of the cross-section of the equal-leg angles, back-to-back type: (

**a**) equal-leg angle; (

**b**) composed section.

**Figure 8.**Optimization of the weight of the prestressed arched truss. (

**a**) Optimization of the weight (in tons) of the prestressed arched truss at different spans; (

**b**) Optimization of the weight (in tons) of the prestressed arched truss at different nodal forces.

**Figure 9.**Comparative analysis of prestressed arched trusses calculated with the classical ($\mathsf{o}$) and optimized (*) methods.

**Table 1.**Geometric dimensions of the equal-leg angles (back-to-back) by GOST 8509–93 (shortened) of the trusses and by Figure 6.

$\mathit{b}$, Width of a Cross-Section, mm | $\mathit{t}$, Thickness, mm | $\mathit{A}$, Area of a Cross-Section, cm^{2} | ${\mathit{i}}_{\mathit{x}}$, Radius of Gyration, cm | ${\mathit{i}}_{\mathit{y}}$, Radius of Gyration, cm | $\mathit{m}$, Mass per Meter, kg/m |
---|---|---|---|---|---|

50 | 5 | 4.88 | 1.53 | 2.45 | 3.77 |

63 | 5 | 6.13 | 1.94 | 2.96 | 4.81 |

70 | 5 | 6.86 | 2.16 | 3.22 | 3.38 |

75 | 6 | 8.78 | 2.30 | 3.44 | 6.89 |

80 | 6 | 9.38 | 2.47 | 3.65 | 7.36 |

90 | 6 | 10.6 | 2.78 | 4.03 | 8.33 |

90 | 7 | 12.3 | 2.77 | 4.06 | 9.64 |

100 | 7 | 13.8 | 3.08 | 4.44 | 10.8 |

100 | 8 | 15.6 | 3.07 | 4.47 | 12.2 |

110 | 8 | 17.2 | 3.39 | 4.87 | 13.5 |

125 | 8 | 19.7 | 3.87 | 5.46 | 15.5 |

125 | 9 | 22.0 | 3.86 | 5.48 | 17.3 |

140 | 9 | 24.7 | 4.34 | 6.09 | 19.4 |

140 | 10 | 27.3 | 4.33 | 6.11 | 21.5 |

160 | 10 | 31.4 | 4.96 | 6.91 | 24.7 |

160 | 11 | 34.4 | 4.95 | 6.93 | 27.0 |

160 | 16 | 49.1 | 4.89 | 7.03 | 38.5 |

180 | 11 | 38.8 | 5.60 | 7.74 | 30.5 |

180 | 12 | 42.2 | 5.59 | 7.76 | 33.1 |

200 | 12 | 47.1 | 6.22 | 8.55 | 37.0 |

200 | 14 | 54.6 | 6.20 | 8.60 | 42.8 |

200 | 16 | 62.0 | 6.17 | 8.64 | 48.7 |

220 | 16 | 68.6 | 6.81 | 9.42 | 53.8 |

250 | 16 | 78.4 | 7.76 | 10.6 | 61.5 |

250 | 20 | 97.0 | 7.71 | 10.7 | 76.1 |

**Table 2.**Steel cable’s geometric, physical and mechanical dimensions by GOST 3068-66 (shortened) for the tie member (cable) of the truss.

Diameter of the Cable, cm | Cross-Section Area, cm^{2} | Mass of the Cable (1 m), kg | Breaking Force of the Cable, kN |
---|---|---|---|

3.8 | 6.61 | 5.86 | 907.5 |

4.2 | 8.15 | 7.24 | 1120 |

4.6 | 9.87 | 8.75 | 1350 |

5.1 | 11.74 | 10.45 | 1610 |

5.5 | 13.81 | 12.25 | 1895 |

5.9 | 16.01 | 14.20 | 2200 |

6.3 | 18.37 | 16.30 | 2525 |

6.8 | 20.90 | 18.55 | 2825 |

Joints | X | Y | Joints | X | Y |
---|---|---|---|---|---|

1 | 0 | 0 | 9 | 15 | 1.5 |

2 | 3 | 2.1 | 10 | 18 | 3.9 |

3 | 6 | 2.7 | 11 | 21 | 3.3 |

4 | 9 | 3.3 | 12 | 24 | 2.7 |

5 | 12 | 3.9 | 13 | 27 | 2.1 |

6 | 15 | 4.5 | 14 | 30 | 0 |

7 | 3 | −0.9 | 15 | 21 | 0.3 |

8 | 9 | 0.3 | 16 | 27 | −0.9 |

Bars | Joint (i) | Joint (j) | Bars | Joint (i) | Joint (j) |
---|---|---|---|---|---|

1 | 1 | 2 | 16 | 6 | 10 |

2 | 2 | 3 | 17 | 10 | 11 |

3 | 3 | 4 | 18 | 11 | 12 |

4 | 4 | 5 | 19 | 12 | 13 |

5 | 5 | 6 | 20 | 13 | 14 |

6 | 1 | 7 | 21 | 9 | 15 |

7 | 7 | 8 | 22 | 15 | 16 |

8 | 8 | 9 | 23 | 16 | 14 |

9 | 7 | 2 | 24 | 15 | 11 |

10 | 8 | 4 | 25 | 16 | 13 |

11 | 9 | 6 | 26 | 9 | 10 |

12 | 7 | 3 | 27 | 10 | 15 |

13 | 3 | 8 | 28 | 15 | 12 |

14 | 8 | 5 | 29 | 12 | 16 |

15 | 5 | 9 | 30 | 1 | 14 |

Span—24 m | Span—27 m | Span—30 m | Span—33 m | |||||||||

Itr. | Weight | P_{0} | Time | Weight | P_{0} | Time | Weight | P_{0} | Time | Weight | P_{0} | Time |

1 | 2.3903 | 200 | 118.553 | 2.3903 | 200 | 118.553 | 2.3903 | 200 | 118.553 | 2.3903 | 200 | 118.553 |

2 | 2.3903 | 200 | 220.806 | 2.3903 | 200 | 220.806 | 2.3903 | 200 | 220.806 | 2.3903 | 200 | 220.806 |

3 | 2.3903 | 200 | 108.674 | 2.3903 | 200 | 108.674 | 2.3903 | 200 | 108.674 | 2.3903 | 200 | 108.674 |

4 | 3.0042 | 890 | 117.505 | 3.0042 | 890 | 117.505 | 3.0042 | 890 | 117.505 | 3.0042 | 890 | 117.505 |

5 | 2.3903 | 200 | 167.809 | 2.3903 | 200 | 167.809 | 2.3903 | 200 | 167.809 | 2.3903 | 200 | 167.809 |

6 | 2.7126 | 570 | 241.135 | 2.7126 | 570 | 241.135 | 2.7126 | 570 | 241.135 | 2.7126 | 570 | 241.135 |

7 | 2.3903 | 200 | 220.899 | 2.3903 | 200 | 220.899 | 2.3903 | 200 | 220.899 | 2.3903 | 200 | 220.899 |

8 | 2.3903 | 200 | 206.569 | 2.3903 | 200 | 206.569 | 2.3903 | 200 | 206.569 | 2.3903 | 200 | 206.569 |

9 | 2.5488 | 380 | 146.296 | 2.5488 | 380 | 146.296 | 2.5488 | 380 | 146.296 | 2.5488 | 380 | 146.296 |

Span—36 m | Span—39 m | Span—42 m | Span—45 m | |||||||||

Itr. | Weight | P_{0} | Time | Weight | P_{0} | Time | Weight | P_{0} | Time | Weight | P_{0} | Time |

1 | 2.3903 | 200 | 118.553 | 2.3903 | 200 | 118.553 | 2.3903 | 200 | 118.553 | - | - | - |

2 | 2.3903 | 200 | 220.806 | 2.3903 | 200 | 220.806 | 2.3903 | 200 | 220.806 | - | - | - |

3 | 2.3903 | 200 | 108.674 | 2.3903 | 200 | 108.674 | 2.3903 | 200 | 108.674 | - | - | - |

4 | 3.0042 | 890 | 117.505 | 3.0042 | 890 | 117.505 | 3.0042 | 890 | 117.505 | - | - | - |

5 | 2.3903 | 200 | 167.809 | 2.3903 | 200 | 167.809 | 2.3903 | 200 | 167.809 | - | - | - |

6 | 2.7126 | 570 | 241.135 | 2.7126 | 570 | 241.135 | 2.7126 | 570 | 241.135 | - | - | - |

7 | 2.3903 | 200 | 220.899 | 2.3903 | 200 | 220.899 | 2.3903 | 200 | 220.899 | - | - | - |

8 | 2.3903 | 200 | 206.569 | 2.3903 | 200 | 206.569 | 2.3903 | 200 | 206.569 | - | - | - |

9 | 2.5488 | 380 | 146.296 | 2.5488 | 380 | 146.296 | 2.5488 | 380 | 146.296 | - | - | - |

_{0}—Prestessing Force (kN); time—(second); Spans—24 m–42 m. Nodal force—10 kN.

Nodal Force—15 kN | Nodal Force—20 kN | Nodal Force—25 kN | Nodal Force—30 kN | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Itr. | Weight | P_{0} | Time | Weight | P_{0} | Time | Weight | P_{0} | Time | Weight | P_{0} | Time |

1 | 4.4736 | 280 | 238.436 | 5.1709 | 440 | 118.123 | 5.8186 | 650 | 113 | 6.4201 | 660 | - |

2 | 4.4832 | 270 | 187.765 | 5.1351 | 380 | 198.251 | 5.8907 | 440 | - | 6.4864 | 650 | - |

3 | 4.4928 | 280 | 270.759 | 5.1231 | 400 | 184.268 | 5.8917 | 430 | - | 6.6117 | 870 | - |

4 | 4.4892 | 280 | 247.101 | 5.1231 | 400 | 167.945 | 5.8186 | 630 | - | 6.4793 | 630 | - |

5 | 4.8831 | 640 | 196.089 | 5.1542 | 430 | 152.751 | 5.8186 | 630 | 124 | 6.6192 | 870 | - |

6 | 4.4736 | 270 | 174.967 | 5.1231 | 400 | 224.255 | 5.9131 | 370 | 164 | 6.4121 | 660 | - |

7 | 4.4867 | 280 | 238.909 | 5.1231 | 400 | 204.561 | 5.8158 | 620 | 116 | 6.6117 | 870 | - |

8 | 4.4747 | 280 | 190.498 | 5.1446 | 370 | 125.209 | 5.8248 | 640 | 115 | 6.4756 | 610 | - |

9 | 4.5094 | 320 | 163.796 | 5.1231 | 400 | 198.452 | 5.8214 | 630 | 132 | 6.4675 | 610 | - |

_{0}—Prestessing Force (kN); time—(seconds); Span of the truss—30 m.

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**MDPI and ACS Style**

Partskhaladze, G.; Alcala, J.; Medzmariashvili, E.; Chavleshvili, G.; Surguladze, B.; Yepes, V.
Heuristic Optimization of a New Type of Prestressed Arched Truss. *Materials* **2022**, *15*, 8144.
https://doi.org/10.3390/ma15228144

**AMA Style**

Partskhaladze G, Alcala J, Medzmariashvili E, Chavleshvili G, Surguladze B, Yepes V.
Heuristic Optimization of a New Type of Prestressed Arched Truss. *Materials*. 2022; 15(22):8144.
https://doi.org/10.3390/ma15228144

**Chicago/Turabian Style**

Partskhaladze, Gaioz, Julian Alcala, Elguja Medzmariashvili, Gocha Chavleshvili, Bichiko Surguladze, and Víctor Yepes.
2022. "Heuristic Optimization of a New Type of Prestressed Arched Truss" *Materials* 15, no. 22: 8144.
https://doi.org/10.3390/ma15228144